Maybe. Presumably, the Google router also provides wifi? Can the old router be set up as a bridge? If so, the setup may or may not be satisfactory. How many floors between upstairs and downstairs? Quite likely the intervening floors and other obstacles to wifi will make it unsatisfactory or unusable. There are just too many variable to be sure until you try it.
You've gone to the expense of having super fast internet access installed, why build in a bottle neck of using wifi equipment to get internet to a desktop computer? This is a situation that "screams" for ethernet wiring. A wiring pro can probably make the connection for not much more than the cost of the equipment you would need to buy to make a wireless system work...